a < key, value > pair, where the value is indexed by the key. Note that keys must ne unique. Consider the example of storing persons name using the social security number (ssn) as the key. For each ssn x, a hash function h is used, where h(x) is the location to store the name of x. Once we have created a table, to look up the name for ssn x, we can recomputes h(x) and then look up what is stored in that location. In Python, dictionaries are based on hash tables. Typically, the hash function h is deterministic; we do not want to get different results every time we compute h(x). But h is often chosen to be pseudo-random. For this problem, we will assume that h is truly random. Suppose there are k people, with each person’s name stored in a random location (independently), represented by an integer between 1 and n, k < n. It may happen that one location has more than one name stored there, if two different people ssns x and y end up with the same random location for their name to

A hash table is a very important data structure in computer science. It is used for fast information retrieval. It
stores data as a < key, value > pair, where the value is indexed by the key. Note that keys must ne unique.
Consider the example of storing persons name using the social security number (ssn) as the key. For each
ssn x, a hash function h is used, where h(x) is the location to store the name of x. Once we have created a
table, to look up the name for ssn x, we can recomputes h(x) and then look up what is stored in that location.
In Python, dictionaries are based on hash tables. Typically, the hash function h is deterministic; we do not
want to get different results every time we compute h(x). But h is often chosen to be pseudo-random. For
this problem, we will assume that h is truly random. Suppose there are k people, with each person’s name
stored in a random location (independently), represented by an integer between 1 and n, k < n. It may
happen that one location has more than one name stored there, if two different people ssns x and y end up
with the same random location for their name to be stored.
1. What is the expected number of locations with no name stored?
2. What is the expected number with exactly one name?
3. What is the expected number with more than one name?